Canonical representations
Govindan
of sp(Zn,R)
Rangarajan
Lawrence BerkeIey Laboratory,
University of California, Berkeley, California 94720
Filippo Neri
AT Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
(Received 5 August 1991; accepted for publication 9 December 1991)
In this paper a rather unconventional real basis for the real symplectic algebra sp(2n,R) is
studied. This basis is valid for representations carried by homogeneous polynomials of
the 2n phase-space variables. The utility of this basis for practical computations is
demonstrated by giving a simple derivation of the second- and fourth-order indices of
irreducible representations of sp ( 2n,R ) .
I. INTRODUCTION
In this paper we study irreducible representations (irreps) of the real symplectic Lie algebra sp ( 2n,R ) ’ carried
by homogeneous polynomials of the 2n phase-space variables {qi,pi} (i = 1,2,...,n). These representations of
sp(2n,R) are the physically interesting ones in classical
mechanics.* Since these are representations carried by
functions of canonically conjugate variables, we refer to
them as canonical representations.
First, we define a real basis for sp(2n,R). Using this
basis, h-reps of sp( 2n,R ) carried by homogeneous polynomials of phase-space variables are obtained. Weight
vectors corresponding to these irreps are also computed.
Using these results, we finally give a simple derivation of
the second- and fourth-order indices corresponding to
these irreps.
II. A REAL BASIS FOR sp(Pn,R)
In this section we obtain a real basis for sp( 2n,R ) .
Since we are interested in representations carried by functions of phase-space variables, it is convenient to define
the operators constituting a basis for sp(2n,R) using
these variables. This can be done using the concept of a
Lie operator.’ Let us denote the collection of 2n phasespace variables qi, pi (i = 1,2,...,n) by the symbol Z. The
Lie operator corresponding to a phase-space function
f(z) is denoted by :f(z) :. It is defined by its action on a
phase-space function g(z) as shown below:
d-(z):g(z)= [ f(z),&) 1.
1
I
*
(2.2)
We are now in a position to give a basis for sp(2n,R).
Consider the following set of real operators:
J. Math. Phys. 33 (4), April 1992
j<k,
:Rjk: = :pipk:,
j<k,
(2.3)
where the indices j and k range from 1 to n. We note that
these are nothing but Lie operators corresponding to the
set of all quadratic monomials in variables qj and pk It
can be shown that these operators constitute a real basis
for the Lie algebra sp(2n,R).3 As expected, there are
n (2n + 1) elements in this basis. We will denote a general basis element by the symbol wf.
The commutator of two Lie operators :f: and :g: can
be shown* to satisfy the following relation:
{:f:,:g:}s:fi:g:
- :g::fi=:[f,g]:.
(2.4)
Using this property, the basis elements are seen to satisfy
the following commutation relations:
(2.5)
(2.1)
Here, cf(z),g(z)]
denotes the usual Poisson bracket of
the functions f(z) and g(z):
[f(z)&(z)]= fi, (~y-~~)
i
I
:Ljk: = :qj q&
Here, the indices j, k, r, and s range from 1 to n.
The basis given in Pq. (2.3) is not the conventional
basis used for sp(2n,R).4 It does not contain a basis for
the unitary algebra u( 3) as a subset. However, this shortcoming is of no consequence when one studies only the
symplectic algebra without any reference to its unitary
subalgebras. In such cases the real basis defined in Eq.
(2.3) offers several advantages due to its one-to-one cor-
f247
Downloaded 04 May 2006 to 159.178.77.96. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
G. Rangarajan and F. Neri: Canonical representations
1248
respondence with the set of quadratic monomials. These
advantages will become apparent in the discussions that
follow.
of sp(217,R)
N(m)
~j-~:P;~‘(z) =
1
d’“‘(f,)~;~‘(z),
/3=1
a= 1,2,..., N(m),
III. REPRESENTATION
OF sp(Sn,R)
In this section we first obtain the irreps of sp(2n,R)
carried by homogeneous polynomials of phase-space variables. Next, we compute the weight vectors corresponding to these irreps.
From the previous section it is clear that Lie operators corresponding to the set of all quadratic polynomials
in z constitute the symplectic Lie algebra sp(2n,R).
Therefore, an N-dimensional representation of sp( 2n,R)
is obtained by mapping each element (denoted in general
by :&:) onto an NXN matrix dV;) such that the following conditions are satisfied for all: f2:, :g2: belonging
to sp(2n,R):
&u-2
d(
[f*,g*l>
+
h?*)
=Mf*)
+
Wg,),
d=@P,
(3.1)
A representation d is said to be (completely) reducible if
it can be converted into a block diagonal form using a
similarity transformation. Otherwise, it is said to be irreducible.
Irreducible representations of sp(2n,R) carried by
homogeneous polynomials in z can be obtained in a
straightforward fashion. Let P”“‘(z)
denote the set of
all homogeneous polynomials in z of degree m and g,(z)
denote a general element belonging to this set. Consider
the following property of the Lie operator :f2: [cf. Eq.
(2.2)]:
:fmn= [fianl
where d’“‘cf2)~ are coefficients multiplying the basis elements.
AswevaryafromltoN(m)inEq.(3.6),thesetof
coefficients dcm)(f2)t gives rise to an N(m) xN(m)
matrix, d’“‘(f,),
for each :f2: belonging to sp(2n,R). It is
easily verified that the set of such matrices (obtained by
letting :f2: range over the entire Lie algebra) gives an
N(m) -dimensional representation of sp (2n,R ) . It can be
shown3 that these representations (for each m) are in fact
irreducible.
Next, we obtain the weight vectors for irreps of
sp(2n,R) defined above. First, we need the Cartan subalgebra of sp( 2n,R). It is easily verified that the Cartan
subalgebra of sp(2n,R) is spanned by the following n
elements:
=h,m
(3.3)
where h, is an element of Y(“)(z).
That is, the elements
:f2: of sp( 2n,R ) preserve the degree of the polynomial on
which they act. We therefore get the following relation:
f*:g,= [ f*,g,]@‘“’
wg&~(~!
(3.4)
In other words, the set of all elements belonging to
sp ( 2n,R ) leaves !ZP(m) invariant.
Let [ PLm’1 be a basis for the set 9 (m). Typically, we
choose these basis elements to be the monomials of degree
m in the 2n phase-space variables. The number of basis
monomials N(m) of degree m in the 2n phase-space variables is given by the relation’V3
H2= :q2p2:, . . . .
H1 = :qlpl:,
(3.2)
=Cebw-2~1.
(3.6)
H,,= :q,,p,,:.
(3.7)
As expected, the rank of sp(2n,R) is equal to n. The
N(m)-dimensional
irrep of Hi is given by Eq. (3.6).
To proceed further we need to choose the N(m) basis
elements PLm’(z). We make the simplest choice. We
choose {PLm’(z)) to be the set of all mth degree monomials in the 2n phase-space variables. Therefore, the general element p’&“)(z) is given by the following relation:
ph)(Z)
cl
=q*~pr2..
.
11
rl +
r211-
lpr2n
q,
r2 +
n ’
a-* + r2n=m.
(3.8)
Now, the advantage of using the rather unconventional basis given in Eq. (2.3) becomes apparent. In this
basis the N( m)-dimensional h-reps of the Cartan basis
elements Hi are automatically diagonal! This is true for
all the irreps defined by Eq. (3.6) Moreover, it is a simple
matter to calculate the general weight vector. Using IQ.
(3.8) we obtain the following results:
:qipi:PLm’(Z)
= (r*i-
r*i-
i= 1,2,...,n.
(3.9)
l)PLm’(Z),
The weight vector A(“’ is defined by the relation
:qip&Pim)(Z)
i= 1,2,...,n.
=A~a)P~m)(Z),
(3.10)
Comparing Eqs. (3.9) and (3.10) we obtain the result
N(m)=(2n+c-1).
(3.5)
From Eq. (3.4) and the completeness of the set {PLm)),
we get the following relation:
Ace)= (r2 -
tor
w4
-
r3,...,r2n
-
r2n-
1).
(3.11)
From Eq. (3.11) we see that the highest weight vecfor the N(m )-dimensional
irrep is given by
J. Math. Phys., Vol. 33, No. 4, April 1992
Downloaded 04 May 2006 to 159.178.77.96. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
G. Rangarajan
and F. Neri: Canonical representations
of sp(Pn,R)
1249
(m,O,...,O). Thus our real basis is valid for the class of
representations with the highest weight vector of the form
(m,O ,..., 0) for m = 1,2,3 ,... .
IV. INDICES OF IRREDUCIBLE
REPRESENTATIONS
Indices of irreducible representation8 play an important part in representation theory. For example, they can
be used to decompose direct products of representations
and compute branching rules. In this section we illustrate
the utility of the basis given in Eq. (2.3) by explicitly
computing the second- and fourth-order indices of the
irreps carried by homogeneous polynomials.
A. Second-order
index
The second-order index is defined as follows:6
I:)=
N(m)
c [P)]2,
Since v(m) is independent of the choice for w, and wa,
we can take both to be equal to qlpl. Then we get the
relation
I(*Ly(m)p*)
m
2
We now turn to the task of computing 1:‘. From Eq.
(4.4) we see that we need to sum over squares of the first
components of the weight vectors corresponding to all the
monomials PLm’(z) (a = 1,2,...N(m)). Since :qlpl: acts
only on the q1 andp, variables, we need to know only the
q1 and p1 content of the basis monomials PLm’(z). Hence,
we write the general monomial as follows:
p(m)(Z)=q~-k-l
a
where
,gl [p]*.
(4.2)
Here, n is the rank of sp(2n,R), 11”) is the ith component
of the ath weight vector, and N(m) is the dimension of
the irrep carried by the mth degree homogeneous polynomials [cf. Eq. (3.5)]. The expression for 1:’ in Eq.
(4.1) can be simplified as follows. From symmetry considerations we have the following relation:
N(m)
C [Aj”‘12=
(4.8)
’
(4.1)
a=1
[A(a)]*=
(4.7)
N(m)
F, [Aja)]*
Vij.
l
Plhk(42,P2,...,4n,Pn),
(4.9)
where hk stands for a monomial of degree k in the 2n - 2
variables q2,p *,...,pn. The index k varies from 0 to m and
the index I from 0 to m - k. The total number of monomials hk of degree k is given by the relation [cf. Eq. (3.5 )]
From Eqs. (3.11) and (4.9) we find the following
expression for the first component Ai”’ of the weight vector corresponding to the monomial PLm’(z) :
(4.3)
a=1
AI”‘=
-m+k+21.
(4.11)
Hence, Eq. (4.1) can be rewritten as follows:
IC)=n
Since all N’(k) monomials with the same value of k have
the same weight, we obtain the result
N(m)
2
[AI”‘]*.
(4.4)
a=1
Before proceeding further,
tion between the second-order
iar Dynkin index.6 Using the
agonal, we obtain the relation
we first establish the relaindex and the more familfact that dcm) (qipi ) is di[cf. Eq. (3.6)]
(4.12)
However, the following identity can be proved:
m-k
&
a= 1,2 ,..., N(m).
However,
relation7
Tr[d’“~(qlp,)d’“~(q,pl)].
the Dynkin
index v(m)
(4.13)
(4.5)
Comparing this with Eq. (3.10) we discover that Eq.
(4.4) can be rewritten as follows:
I(*)=.
m
(;I;+;).
(-m+k+202=2
Therefore, we obtain the relation
1:‘=2n
TI,I
C
(‘“+;-y:+y).
(4.14)
(4.6)
is defined by the
After some manipulation,
following result:
this can be shown to equal the
J. Math. Phys., Vol. 33, No. 4, April 1992
Downloaded 04 May 2006 to 159.178.77.96. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
1250
G. Rangarajan and F. Neri: Canonical representations
I(‘
=2n2n2n+l+ m ’
m)
( 1
Similarly,
(4.15)
It is easily verified that this expression gives the correct
second-order indices* for the irreps being studied (apart
from an overall numerical factor).
From Eq. (4.8) the Dynkin index v(m) is given by
the relation
v(m) =
2n + m
2n+ 1 (2n+2)-‘.
(
)
N(m)
z, (~~a92(~:a92
m-l
=
m-k
kzo
&
( -
m +
k +
202
(4.23)
This can be simplified to the following
expression:
(4.17)
C2W =N(2MmVN(m),
where N(2) is equal to the dimension of sp (2n,R). We
obtain the following result:
C,(A) = (m2 + 2nm)/4(n + 1).
=2~~,’
rL!y2)
y$k(-m+k+21)i.
(4.24)
(4.18)
Evaluating this double sum, we obtain the result
index
The fourth-order
we can derive the relation
(4.16)
Further, the eigenvalue C,(A)
of the second-order
Casimir operator can be calculated using the following
relation:7
8. Fourth-order
of sp(2n,R)
index is defined as follows:6
(4.25)
I:)=
IT,)
[ i,
cni”‘,‘]‘.
(4.19)
Using symmetry considerations this can be rewritten as
follows:
N(m)
IE)=n
C
(Ai”‘)4+
n(n - 1)
N(m)
2,
(4.20)
(~Ia92(w)2.
2n + m
Ic4)
m =2n2n + 1 (2mn2 + m2n + IOmn + 5m2
( 1
- 6n - 2)/(2n + 3)(n + 1).
a=1
x
Substituting Eqs. (4.22) and (4.25) in Eq. (4.20), we
finally get the following relation:
Proceeding as before we lind, after some manipulation, the following relation:
(4.26)
Again, it is easily verified that the above expression for
1:’ gives correct results (up to an overall numerical factar). Comparing Eqs. (4.15) and (4.26) we also obtain
the relation
1g)=1E)(2mn2 + m2n + 10mn + 5m2 - 6n - 2)
x[(2n+3)(n+l)lp1.
X
(4.27)
[ - 8 + 12(m -k)
+ 6(m - @l/5.
(4.21)
Evaluating the sums, we obtain the result
(2n2 + 6m2 + 12mn - 7n
- 3)/(2n
+ 3)(n + 1).
(4.22)
V. SUMMARY
In this paper we discuss a real basis for sp (2n,R ) . In
this basis the Cartan basis elements are automatically diagonal for all h-reps carried by homogeneous polynomials
of phase-space variables. This fact facilitates the computation of various quantities characterizing these irreps. As
an illustration, we calculate the second- and fourth-order
indices of these h-reps.
J. Math. Phys., Vol. 33, No. 4, April 1992
Downloaded 04 May 2006 to 159.178.77.96. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
G. Rangarajan
and F. Neri: Canonical representations
ACKNOWLEDGMENTS
One of us (GR) was supported by the United States
Department of Energy under contract number DEACQ3-76SFF00098.
We would like to thank Dr. Alex Dragt for useful
discussions.
‘The reader should be warned that there are several conflicting notations in use for denoting the various symplectic algebras. The notation
sp( 2n) is oRen used to denote the real symplectic algebra sp( 2n,R).
Sometimes sp(n,R) is used to denote this algebra. In this paper we
have adopted the least ambiguous set of notations found in the literature. The real symplectic algebra is denoted by sp(2n,R); the complex symplectic algebra is denoted by sp(2n,C); the so-called unitary
symplectic algebra is denoted by sp(2n).
of sp(2n,R)
1251
‘A. J. Dragt, in Physics of High Energy Particle Accelerators, edited by
R. A. Carrigan, F. R. Huson, and M. Month, AIP Conference Proceedings No. 87 (American Institute of Physics, New York, 1982), p.
147; see also A. J. Dragt et aL, Ann. Rev. Nucl. Part. Sci. 38, 455
(1988).
“G. Rangarajan, Ph. D. thesis, University of Maryland, 1990.
4H. Georgi, Lie Algebras in Particle Physics (Benjamin/Cummings,
Reading, MA, 1982).
s A. Giorgilli Comp. Phys. Comm. 16, 331 (1979).
6J. Patera, R. T. Sharp, and P. Wintemitz, 3. Math. Phys. 17, 1972
(1976); J. Patera, R. T. Sharp, and P. Wintemitz, J. Math. Phys. 18,
1519 (1977).
‘J. F. Comwell, Group Theory in Physics (Academic, London, 1984),
Vols. 1 and 2.
s W. G. McKay and J. Patera, Tables of Dimensions, Indices, and
Branching Rules for Representations of Simple Lie Groups (Marcel
Dekker, New York, 1981).
J. Math. Phys., Vol. 33, No. 4, April 1992
Downloaded 04 May 2006 to 159.178.77.96. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp